Induced lines in Hales–Jewett cubes

A line in [ n ] d is a set { x ( 1 ) , … , x ( n ) } of n elements of [ n ] d such that for each 1 ⩽ i ⩽ d , the sequence x i ( 1 ) , … , x i ( n ) is either strictly increasing from 1 to n, or strictly decreasing from n to 1, or constant. How many lines can a set S ⊆ [ n ] d of a given size contain? our aims in this paper is to give a counterexample to the Ratio Conjecture of Patashnik, which states that the greatest average degree is attained when S = [ n ] d . Our other main aim is to prove the result (which would have been strongly suggested by the Ratio Conjecture) that the number of lines contained in S is at most | S | 2 − ε for some ε > 0 . o prove similar results for combinatorial, or Hales–Jewett, lines, i.e. lines such that only strictly increasing or constant sequences are allowed.